Recently I've dived into studying complex analysis on my own and it led me to find an interesting way of displaying the Mandelbrot set:
Consider $$f(f(f(f(...x))))$$ Then, $$ \frac{d}{dx} (f(f(f(f(...x))))) = f'(f(f(f(...x))))\cdot f'(f(f(...x)))...$$ Assuming that this function converges for $x$, $$\frac{d}{dx}(f(f(...x))) = \prod_{p=1}^{\infty} f'(f^{p}(x))$$ Since all non-edge-points in a Julia set always diverge or converge to the same cycle, this gave me an idea. Even though this product is always equal to $0$ or $\infty$, I wondered if it was possible to still compare the derivatives by how quickly they converge to $0$. I thought about the best way to do this for a while and came up with the following expression to color the points in the Mandelbrot set: $$ 1-4\sqrt[p]{ \prod_{i=1}^{p} d(v_{i}) }$$ where $p$ is the length of the cycle the point converges to, $v_i$ is the $i$th point of the cycle, and $d(x)$ is a function which returns the squared distance of $x$ from the origin. The radical normalizes the value of the product based on the length of the cycle, in effect determining the "average" value of $f'(f^p(x))$. Finding the cycle and its values can be done efficiently using pre-existing algorithms. When I made a program to implement this algorithm, I got the following really cool image:
If this has been done before, what is this method of displaying the set called?